Thanks for your question. I don't doubt for a second that it is well intentioned, but much of what your are asking is, unfortunately, poorly phrased. The line of questioning has two major flaws that we should start with.

First, the concept of “races” is problematic because different people use the word to mean very different ideas depending on a whole variety of things. There is no such things as a biological or “objective" race, so the various concepts people refer to when they talk about races is complicated and often very confusing. See these AskAnthropology threads for more info: Race is a social construct and on Prehistoric "races". They know their stuff over there.

Second, your question conflates perceptions of identities (“considered their own”) with expressions of identities (“genetic characteristics”). The field of anthropology has whole sub-fields dedicated to the clarification of the differences between how groups understand themselves versus how other see them. Identity is a very tricky thing. When we combine this issue with the first issue, it is very difficult to get a group of people to agree that we are all talking about the same idea of difference.

It was very helpful that you tried to define what you mean by “race.” Any serious discussion about the topic requires a definition to get everyone on the same page, but a “large group of people” is, unfortunately, too vague to be really helpful. Judging from the list of examples you give, I'm going to try to start answering some of what I think you are talking about, but take the clarifications I mentioned above seriously.

When you say, “By race I mean large groups of people that are visibly distinct from each other I.e. Indians, Arabs, Asians, caucasians, Africans , etc.” and "How many races on earth today existed in the same form in antiquity ?”

I think you are really asking about what anthropologists call Primary Civilizations or Pristine Cultures. These are independent communities that developed into “complex societies” without inspiration from outsiders. The traditional list of such societies is six: Egypt, Mesopotamia (Sumer), Mesoamerica (Maya/Aztec) South America (Inca), China, and Indus Valley (Harappa), but some scholars include a few others.

Would the people who inhabited Mesopotamia be considered their own distinct race or is there a race alive today that has more or less the same genetic characteristics?

The Sumerians, Assyrians, and Babylonians did not have a concept of race. They understood the world to be of one human species that was divided into various “peoples” according to the place and manner in which they lived. Naturally, they imagined that “they” were superior to the others, which meant that anyone who lived in a city and worshipped gods in city-temples and owned a lot of property was better than anyone else. Anyone could move to the city, worship the right gods, and basically "become" Babylonian. That's the main narrative of Enkidu in the Epic of Gilgamesh.

Are their entire races that have died out?

There were entire humanoid species that have died out. It's not my area of expertise, but you should definitely look into hominid research if you want to hear about extinction events.

As for civilizations, there are many non-literate civilizations that ended before leaving writing that could be studied. Pre-Columbian American “history” is full of such societies, but you could also look at places like the Old Europe or the Capsian Culture if you want to see entire cultures that have basically vanished.

Was racial diversity higher in the past?

For the reasons I mentioned above, it's impossible to answer a question like this, unfortunately. What boundaries should we use to define “racial” diversity?

There has probably only ever been up to six independent world-systems (i.e. large “civilizations”) functioning on the earth, but each of them contained various social and cultural ways of understanding difference within and without independent societies.

One could use language as a way of evaluating diversity. There are somewhere between 1000 and 6500 languages in the world today, in an era of unprecedented unification and standardization. It was many times more in the past. But if language is any indication of diversity, then we are heading into an era of unprecedented similarity. Languages are steadily disappearing from the planet.

Now this is a tough question to answer well, so I'll take a stab as one of AH's resident anthropologists. Let's go through it point by point.

Race is not biologically defined.: The social category of "race" is a conglomeration of various phenotypes (physical expressions of genes), ethnicity, socio-economic status, political census categories and the like. Different elements have been used to distinguish races in different cultures across time, recently in the West it's just happened to be skin color. You've approached this issue by providing your own definition, which gives you 5000 non-sarcastic brownie points, but it doesn't quite solve it.

That means we can't group past people into present races. We can and do group archaeological populations into groups on ethnicity based on where they came from, or hereditary/familial groups. But this does not translate into race. Race is a social category, whose utility is in understanding relations of a particular period. It does not transcend time (more specifically, societal change).

Humans have interbred too much in the past 6000 years to section off races by continual heredity. Races being "visibly distinct" is not a practical way to categorize past populations. People argue all the time about whether King Tut was Black or White; truth is, he was Egyptian, and we can't determine the skin color of a 3000 year old mummy. Now what about me? My ancestors are primarily German, Welsh, and Cherokee. My skin is white, but many of my skeletal (and a few external) phenotypes are stereotypical of native populations in the SE US. They dig up my bones 3000 years from now, whose to say I'm Caucasian. I am, since that what I identify is, but if they're gonna guess from my bones and DNA, they could very well get it wrong.

There's more genetic diversity within races than without. Last point, pretty straightforward.

You'll see your liberal wannabe-activist friends say on Facebook that race doesn't exist. This is wrong. Race exists very strongly as a socially defined category. What they mean to say is that race has no objective or academic definition. You will no see academics categorize people by race except when A) people are defining themselves or B) they are studying the social definition itself. It is simply not a category we use.

So to give some answers:

Would Mesopotamian be a distinct race? Well, it would be wrong to categorize them as any present race. And, as said before, to assign a race based on genetics isn't what scholars do.

Have races died out? Again, this relies on strict categorization of race by lineage. Substantial groups of people have been killed, yes, but there is no official statement of "yes, they were a race."

Was racial diversity higher? It is terrible difficult to measure genetic diversity. However, the sheer population increase sense even 500 AD has certainly increased the gene pool.

How many present races existed back then? Yet again, we cannot categorize the past's people by today's social categories. There would have been people you would call "Asian" based on their features. But it would have no meaning or purpose without an "Asia" and a "not Asia." Putting them into some arbitary "race" really only gets in the way of good discussion.

I hope that all makes sense. I'm definitely open to more clarifying question. I hate to completely toss aside your question, but the truth is that "race" has not been a concept used by academics for ages. Categorizing the people of the earth into well-defined races based on appearance, or even genetics, has gone the way of phlogiston.

Is there any evidence for other human species surviving into the time period in question?